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Physics-Informed Global Extraction of the Universal Small-$x$ Dipole Amplitude

Si-Wei Dai, Fu-Peng Li, Long-Gang Pang, Guang-You Qin, Shu-Yi Wei, Han-Zhong Zhang, Wenbin Zhao

Abstract

We extract the universal small-$x$ dipole scattering amplitude $N(r,x_B)$ from a global analysis based on a physics-informed neural network (PINN), without imposing a priori MV-type parametrization of the initial condition. The network provides a smooth and differentiable surrogate for $N(r,x_B)$, whose rapidity dependence is constrained by the collinearly improved Balitsky--Kovchegov evolution equation, while its functional form is simultaneously constrained by Deep Inelastic Scattering (DIS) data for the reduced total and charm cross sections, exclusive $J/ψ$ photoproduction measurements, and a positivity requirement for the momentum-space dipole amplitude. The resulting single universal amplitude consistently describes all fitted observables within a unified framework, alleviating the long-standing tension between total and charm channels encountered in conventional small-$x$ fits based on rigid parametric ansätze. Within the fitted kinematic domain, the best extracted PINN solution yields a smooth, non-negative momentum-space dipole over the full transverse-momentum range examined. Our results provide a robust and well-behaved input for Color Glass Condensate phenomenology across a broad class of high-energy processes.

Physics-Informed Global Extraction of the Universal Small-$x$ Dipole Amplitude

Abstract

We extract the universal small- dipole scattering amplitude from a global analysis based on a physics-informed neural network (PINN), without imposing a priori MV-type parametrization of the initial condition. The network provides a smooth and differentiable surrogate for , whose rapidity dependence is constrained by the collinearly improved Balitsky--Kovchegov evolution equation, while its functional form is simultaneously constrained by Deep Inelastic Scattering (DIS) data for the reduced total and charm cross sections, exclusive photoproduction measurements, and a positivity requirement for the momentum-space dipole amplitude. The resulting single universal amplitude consistently describes all fitted observables within a unified framework, alleviating the long-standing tension between total and charm channels encountered in conventional small- fits based on rigid parametric ansätze. Within the fitted kinematic domain, the best extracted PINN solution yields a smooth, non-negative momentum-space dipole over the full transverse-momentum range examined. Our results provide a robust and well-behaved input for Color Glass Condensate phenomenology across a broad class of high-energy processes.
Paper Structure (6 sections, 21 equations, 7 figures)

This paper contains 6 sections, 21 equations, 7 figures.

Table of Contents

  1. SUPPLEMENTAL MATERIAL
  2. A. Physics-Informed Neural Network Framework and Loss Function
  3. B. Theoretical Formalism for the Observables and Supplementary Results
  4. B1. Reduced cross sections and longitudinal structure function: $\sigma_r$, $F_L$ and $\sigma_r^{c\bar{c}}$
  5. B2. Diffractive Vector-Meson Production
  6. B3. Reduced cross sections and longitudinal structure function at all energies

Figures (7)

  • Figure 1: The reduced total and charm cross sections, $\sigma_r$ and $\sigma_r^{c\bar{c}}$, are shown vs. $x_B$ in selected $Q^2$ bins at $\sqrt{s}=318~\mathrm{GeV}$, compared with HERA data H1:2015ubcH1:2009pzeH1:2018flt. Solid curves are the best-fit PINN prediction (minimum global $\chi^2/\mathrm{d.o.f.}$); bands show $1\sigma$ uncertainties. The quoted $\chi^2/\mathrm{d.o.f.}$ uses the solid curve; for $\sigma_r$ it includes all available energies.
  • Figure 2: Total cross section for exclusive $J/\psi$ photoproduction as a function of center-of-mass energy $W$, compare to ALICE ALICE:2014eofALICE:2018oyo, H1 H1:2005dtpH1:2013okq, ZEUS ZEUS:2002wfj and LHCb LHCb:2018rcmLHCb:2024pcz data. Solid curve is the PINN prediction with the smallest global $\chi^2/\mathrm{d.o.f.}$; shaded band is $1\sigma$ uncertainties. The quoted $\chi^2/\mathrm{d.o.f.}$ value is evaluated using the solid curve.
  • Figure 3: Top: Solid curves show the dipole amplitude $N(r,x_B)$ predicted by the PINN for $x_B \in \{10^{-2},\,10^{-3},\,10^{-5}\}$. Markers denote numerical ciBK evolution results obtained with the same initial condition as used in the PINN. The dashed curve corresponds to the best-fit MV-type parametrization to $N_{\rm PINN}(r,x_B=0.01)$.
  • Figure 4: Fourier transform of the dipole amplitude at $x_B \in \{10^{-2},\ 10^{-3},\ 10^{-5}\}$.
  • Figure S1: Schematic of the PINN framework. The network maps the kinematic inputs $(r,Y)$ to the dipole amplitude $N(r,Y)$. Training is guided by the total loss $\mathcal{L}(\theta,\mathbf{p})=w_1\mathcal{L}_{\text{ciBK}}+w_2\mathcal{L}_{\text{data}}+w_3\mathcal{L}_{\text{phy}}.$ See the text for details of the three loss components.
  • ...and 2 more figures